I am studying some particular examples of hyperelliptic curves and their automorphism groups (from this paper, if it is of interest).
It is mentioned in the paper to understand the automorphism of some curves it is necessary to rewrite the defining equations such that only even powers of $x$ occur; i.e. find a model for the curve of the form $$ y^2 = \sum_{i=0}^s a_{2i}x^{2i}. \qquad \qquad (1) $$
The paper does not specify how one should do this, and I am struggling to do this with some of the curves given. I imagine there may be some techniques I am missing, so I wondered if anyone could help with one of the examples.
We suppose that we are working over an algebraically closed field $k$ of characteristic three. The model for the modular curve $X_0(50)$ given in the aforementioned paper is \begin{equation} y^2 = x^6 + 2x^5 + 2x^3 + 2x + 1. \end{equation} My problem seems to arise from the fact I can't remove the $x^5$ term when trying to write this in the form of $(1)$. The only automorphism I have needed to use in previous examples has been $x \mapsto x+c$ for some constant $c \in k$. However, since $(x+c)^6 = x^6 + 2c^3x^3 + c^6$ in characteristic three this clearly won't work. Are there more subtle/less obvious methods I could use? Or is it not possible to find a model of the form $(1)$ in this manner?
Thanks in advance for any help, and please let me know if I need to clarify anything.
Perhaps a useful observation is that a curve in the form $(1)$ has two distinct automorphisms of order two, namely $x\mapsto -x$ and $y \mapsto -y$. I do not think that a generic hyperelliptic curve has two such automorphisms. The automorphism $y\mapsto -y$ always exists (on your modular curve it corresponds to the Atkin-Lehner involution) but the existence of another automorphism of order $2$ is not so clear.